Weak-Field (Anomalous) Zeeman Effect

A year after Zeeman's triplets, Thomas Preston photographed magnetic splittings that should not have existed: sodium lines breaking into four and six components, with spacings in strange rational ratios. Classical theory had exactly one prediction — three lines — and nature was cheerfully ignoring it. For twenty-eight years the "anomalous Zeeman effect" was atomic theory's most public embarrassment; Pauli described being unable to look happy while thinking about it. The resolution required a new degree of freedom (spin), a new coupling rule (the $j$ -ladder of fine structure), and one projection argument. This article is the payoff side of that story. The machinery is all inherited: the Zeeman Hamiltonian and the regime taxonomy come from the previous serial (Zeeman Effect), the legality of first-order theory in the coupled basis comes from Good Quantum Numbers and Symmetry-Adapted Bases, and the central technical result — the Landé factor $g_J$ , evaluated by projecting $\hat{\mathbf{L}}+2\hat{\mathbf{S}}$ onto $\hat{\mathbf{J}}$ — was derived in Projection Theorem for Vector Operators and is consumed here, not re-proved. What this article owns is the application anatomy: each fine-structure level fans into $2j+1$ equally spaced rungs with its own spacing $g_J\mu_B B$ , and because upper and lower levels of a transition fan at different rates, spectral lines split into the quartets and sextets that baffled the 1890s. We compute the full component census for hydrogen's Lyman-α and sodium's D lines — four and six components, every energy machine-verified — and derive honestly where the weak-field description itself breaks: at what field the neighboring $j$ -level starts leaking in.

Key takeaways

Regime (cashed from Zeeman Effect): $\mu_B B\ll\Delta E_{fs}$ — fine structure diagonalizes first, and the field perturbs the $|n\,\ell\,j\,m_j\rangle$ ladder; $j,m_j$ are the good labels per Good Quantum Numbers and Symmetry-Adapted Bases.
First order: $E^{(1)}_Z=g_J\,\mu_B B\,m_j$ with $g_J=1+\dfrac{j(j+1)+s(s+1)-\ell(\ell+1)}{2j(j+1)}$ — the Landé factor, derived in Projection Theorem for Vector Operators and consumed here.
Each level fans into $2j+1$ equally spaced rungs; different levels fan at different rates: $g(1s_{1/2})=2$ , $g(2p_{1/2})=\tfrac23$ , $g(2p_{3/2})=\tfrac43$ (all recomputed).
Lines are differences of two fans: with $\Delta m_j=0,\pm1$ , hydrogen Lyman-α splits into $4+6$ components and sodium D₁/D₂ into 4 and 6 — the anomalous patterns, counted exactly (programmatic sweep).
All shifts are rational multiples of $\mu_B B$ (thirds, for these lines) — Runge's 1907 rule, now a two-line consequence of $g_J$ arithmetic.
Validity edge computed: $j$ -mixing between $2p_{1/2}$ and $2p_{3/2}$ reaches 10% at $B\approx0.17$ T — "weak field" for hydrogen $n=2$ ends well below a tesla.
Key idea: in a weak field each fine-structure level is a tiny equally-spaced ladder with its own Landé spacing $g_J\mu_B B$ ; every anomalous pattern is just the difference table of two such ladders under $\Delta m_j=0,\pm1$ .

The core insight

In a weak field the atom keeps its fine-structure identity: states are $|n\,\ell\,j\,m_j\rangle$ , and the field merely fans each $j$ -level into $2j+1$ rungs spaced $g_J\mu_B B$ apart. Because $g_J$ differs level by level, transition patterns are anomalous — richer than the classical triplet — and fully predicted by one factor consumed from the projection theorem.

Where this lives today

The weak-field Zeeman fan is the working interface between atoms and magnetic technology. Optical pumping drives σ⁺ transitions up one fan rung at a time to spin-polarize entire vapors — the mechanism inside every alkali magnetometer and the hyperpolarizers that light up noble-gas MRI. Atomic clocks operate between fans: the clock transition connects $m=0$ states with no linear shift, while the $m\ne0$ rungs on either side are monitored as built-in gauges of stray field — every cesium fountain runs this article's arithmetic (one coupling deeper, with $g_F$ replacing $g_J$ via the same projection logic) thousands of times a second. And in astrophysics, measured $g_J$ -weighted splittings identify not just field strengths but which term produced a line, making the Landé factor a fingerprint reader for stellar spectra — spectral-line catalogs list $g_J$ next to wavelength for exactly this purpose.

The physical picture

The previous serial's tilted-moment picture does the work. Spin-orbit coupling locks $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ into rapid precession about $\hat{\mathbf{J}}$ ; a weak field only sees the time-averaged moment — the component along $\hat{\mathbf{J}}$ . The average of $\hat{\mathbf{L}}+2\hat{\mathbf{S}}$ along $\hat{\mathbf{J}}$ is $g_J\hat{\mathbf{J}}$ , so the level responds as if it had a single effective moment $g_J\mu_B$ per unit of $m_j$ . All the anomalous richness is in that one number being different for every $(\ell,j)$ pair.

Figure

Each fine-structure level fans at its own rate: the three ladders of the Lyman-α system at 1 T

Now put two fans under one transition. The photon energy is (line center) + (upper rung) − (lower rung). If both fans had the same spacing, differences would collapse to three values — the normal triplet of the previous serial. But $g(2p_{3/2})=\tfrac43$ while $g(1s_{1/2})=2$ : the difference table refuses to collapse, and Lyman-α's blue component set spreads into six distinct energies. That is the entire anomaly. Nothing is anomalous in the mechanics; only the 1890s expectation was.

Figure

Anomalous pattern anatomy: the six components of the 2p₃/₂ → 1s₁/₂ line

The shift formula, assembled from owned parts

Everything hinges on evaluating $\hat{H}_Z=(\mu_B B/\hbar)(\hat{J}_z+\hat{S}_z)$ — the form organized by the previous serial — inside a fine-structure eigenstate. The regime premise ( $\mu_B B\ll\Delta E_{fs}$ , the weak column of Zeeman Effect's taxonomy) makes $|n\,\ell\,j\,m_j\rangle$ the unperturbed states, and the good-states theorem of Good Quantum Numbers and Symmetry-Adapted Bases certifies first-order legality: $\hat{H}_Z$ commutes with $\hat{J}_z$ (field along the quantization axis), so within a fixed- $j$ multiplet the perturbation is already diagonal in $m_j$ — no secular matrix needed, the "fortunate basis" case of Degenerate Perturbation Theory.

E^{(1)}_Z \;=\; \frac{\mu_B B}{\hbar}\,\big\langle n\,\ell\,j\,m_j\big|\,\hat{J}_z+\hat{S}_z\,\big|n\,\ell\,j\,m_j\big\rangle \;=\; \mu_B B\,m_j\;+\;\frac{\mu_B B}{\hbar}\,\langle\hat{S}_z\rangle \qquad (1)

where: the $\hat{J}_z$ term is exact and trivial ( $\hbar m_j$ ); the entire problem is $\langle\hat{S}_z\rangle$ — the average spin projection inside a state where spin is locked to orbit. This is precisely the question Projection Theorem for Vector Operators was built to answer: within a fixed- $j$ multiplet every vector operator acts as its projection along $\hat{\mathbf{J}}$ , giving $\langle\hat{S}_z\rangle=\big[\langle\hat{\mathbf{S}}\cdot\hat{\mathbf{J}}\rangle/\hbar^2j(j+1)\big]\hbar m_j$ , and its Landé derivation lands on

E^{(1)}_Z \;=\; g_J\,\mu_B B\,m_j\,,\qquad g_J \;=\; 1+\frac{j(j+1)+s(s+1)-\ell(\ell+1)}{2\,j(j+1)} \qquad (2)

where: $g_J$ is the Landé g-factor — consumed from that article, not re-derived here, together with its five verified special cases and its two-route consistency check. Read the structure: the "1" is the orbital baseline (a $g_s=1$ world would stop there), and the fraction is spin's excess wage, positive when spin is aligned with $\hat{\mathbf{J}}$ ( $j=\ell+\tfrac12$ ) and negative when anti-aligned ( $j=\ell-\tfrac12$ , where $g_J$ dips below 1 — the $\tfrac23$ of $2p_{1/2}$ ). The three values this article's patterns need, each recomputed from (2): $g(1s_{1/2})=2$ , $g(2p_{1/2})=\tfrac23$ , $g(2p_{3/2})=\tfrac43$ . The fan anatomy follows immediately and is worth stating as a card of its own:

\text{level } (n,\ell,j)\ \longrightarrow\ 2j+1 \text{ rungs},\quad \text{spacing } g_J\,\mu_B B,\quad \text{centroid unshifted} \qquad (3)

where: equal spacing holds because (2) is linear in $m_j$ ; the centroid stays put because $\sum_{m_j}m_j=0$ — the field redistributes a multiplet's energy but does not move its center of gravity, the same sum-rule logic as the spin-orbit centroid theorem in Spin-Orbit Coupling. A $j=\tfrac32$ level becomes a quartet spanning $3g_J\mu_B B$ ; a $j=\tfrac12$ level a doublet spanning $g_J\mu_B B$ .

Derivation AThe weak-field shift, with the projection theorem as a cited input

We assemble $E^{(1)}_Z=g_J\mu_B Bm_j$ from owned parts, being explicit about what is consumed and what is checked. The strategy: verify legality (why plain first-order theory is allowed inside a degenerate multiplet), split the operator into its trivial and nontrivial pieces, and hand the nontrivial piece to the projection theorem — Projection Theorem for Vector Operators proved it and derived $g_J$ ; we import the result and audit only the arithmetic of the values we use. Watch the legality step: it is the article's only genuinely delicate point.

Step 1. Legality. The $2j+1$ states of a fine-structure level are degenerate, so Degenerate Perturbation Theory demands we diagonalize $\hat{H}_Z$ within the multiplet. But $[\hat{H}_Z,\hat{J}_z]=0$ for $\mathbf{B}\parallel\hat z$ , and the multiplet's states carry distinct $\hat{J}_z$ eigenvalues — exactly the good-operator situation of Good Quantum Numbers and Symmetry-Adapted Bases: the $W$ -matrix is automatically diagonal in the $m_j$ basis, and first-order shifts are plain expectation values.

Step 2. Split. $\hat{L}_z+2\hat{S}_z=\hat{J}_z+\hat{S}_z$ (the previous serial's organizing identity). The $\hat{J}_z$ piece contributes $\mu_B Bm_j$ exactly.

Step 3. Project. Inside $|n\,\ell\,j\,m_j\rangle$ , spin has no independent life: it precesses about $\hat{\mathbf{J}}$ , and only its $\hat{\mathbf{J}}$ -component survives the average. Formally — the projection theorem, consumed: $\langle\hat{S}_z\rangle=\dfrac{\langle\hat{\mathbf{S}}\cdot\hat{\mathbf{J}}\rangle}{\hbar^2 j(j+1)}\,\hbar m_j$ with $\langle\hat{\mathbf{S}}\cdot\hat{\mathbf{J}}\rangle=\tfrac{\hbar^2}{2}[j(j+1)+s(s+1)-\ell(\ell+1)]$ . Substituting gives (2); the derivation, the $\langle\hat{S}_z\rangle=\pm m\hbar/(2\ell+1)$ table, and the five special-case audits are that article's property. Why the projection is legitimate only within one j: the theorem says nothing about matrix elements between different j — those are generically nonzero and become this article's breakdown mechanism in Derivation C. Weak-field theory is the approximation that they haven't yet been given the energy leverage to matter.

Step 4. Audit the used values. From (2): $1s_{1/2}$ ( $\ell=0$ ): $g=2$ ; $2p_{1/2}$ : $g=1-\tfrac13=\tfrac23$ ; $2p_{3/2}$ : $g=1+\tfrac13=\tfrac43$ . Sanity limits: $s=0\Rightarrow g_J=1$ (normal triplet of Zeeman Effect recovered — consistency across the serial boundary), $\ell=0\Rightarrow g_J=2$ (pure spin, the moment dictionary's $g_s$ ). Verified: g-values 2, 2/3, 4/3 recomputed from (2); both symbolic limits (s=0 ⇒ 1, ℓ=0 ⇒ 2) by sympy; the 0931-owned sodium anchors (4/3)×57.88 = 77.2 μeV and (2/3)×57.88 = 38.6 μeV re-checked as arithmetic (article verify script).

The anomalous patterns, counted exactly

A spectral line connects two fans. With upper rung $g_um_u$ and lower rung $g_lm_l$ (in units of $\mu_B B$ ), the photon shift is $g_um_u-g_lm_l$ , pruned by the electric-dipole rule $\Delta m_j=0,\pm1$ (stated here; its derivation belongs to the time-dependent chapter). Because $g_u\ne g_l$ , the difference table does not telescope — the count and the spacings become fingerprints of the two levels' quantum numbers.

h\Delta\nu \;=\; \mu_B B\,\big(g_u\,m_u-g_l\,m_l\big)\,,\qquad m_u-m_l\in\{0,\pm1\} \qquad (4)

where: each allowed $(m_u,m_l)$ pair is one component; $\Delta m_j=0$ components are π-polarized (dark along $\mathbf{B}$ ), $\Delta m_j=\pm1$ are σ± — the polarization dictionary carried over intact from Zeeman Effect. Equation (4) is the entire anomalous Zeeman effect: everything else is enumeration, which is best done honestly, by sweep.

Go deeper · Derivation B · 2 minComponent census for Lyman-alpha and the sodium D lines

Derivation BComponent census for Lyman-alpha and the sodium D lines

We enumerate the components of the two classic anomalous systems — hydrogen Lyman-α ( $2p_{1/2},2p_{3/2}\to1s_{1/2}$ ) and sodium D ( $3p_{1/2},3p_{3/2}\to3s_{1/2}$ ), which share the same $(j,g)$ structure — and prove the counts 4 and 6 with every shift evaluated. The strategy is a programmatic sweep of (4) over all allowed pairs; the payoff is Runge's 1907 observation, mysterious for two decades, falling out as trivial arithmetic: all shifts are integer multiples of $\tfrac13\mu_B B$ .

Step 1. The $j=\tfrac12\to j=\tfrac12$ line (D₁-type; hydrogen $2p_{1/2}\to1s_{1/2}$ ). Upper rungs $g_um_u\in\{\pm\tfrac13\}$ ( $g_u=\tfrac23$ ), lower rungs $g_lm_l\in\{\pm1\}$ ( $g_l=2$ ). All four $(m_u,m_l)$ combinations satisfy $|\Delta m_j|\le1$ , giving shifts $\pm\tfrac13\mp1$ and $\pm\tfrac13\pm1$ : the set $\{\pm\tfrac23,\pm\tfrac43\}$ — four components, no unshifted line. The π pair ( $\Delta m=0$ ) sits at $\mp\tfrac23$ ; the σ pair at $\pm\tfrac43$ .

Step 2. The $j=\tfrac32\to j=\tfrac12$ line (D₂-type; hydrogen $2p_{3/2}\to1s_{1/2}$ ). Upper rungs $\{\pm2,\pm\tfrac23\}$ ( $g_u=\tfrac43$ ), lower $\{\pm1\}$ . The rule kills $m_u=\pm\tfrac32\to m_l=\mp\tfrac12$ ; six pairs survive with shifts $\{\pm\tfrac13,\pm1,\pm\tfrac53\}$ — six components: π pair at $\pm\tfrac13$ , σ quartet at $\pm1,\pm\tfrac53$ . Why no component sits at zero: an unshifted line needs g_u m_u = g_l m_l, impossible here because the g's are unequal and the m's are half-integers — the visible signature that both levels carry spin. A singlet system, by contrast, always keeps its central π line.

Step 3. Runge's rule, demystified. Every shift in Steps 1-2 is $k/3$ in units of $\mu_B B$ , $k\in\{\pm1,\pm2,\pm3,\pm4,\pm5\}$ (swept programmatically). The denominator 3 is $2j(j+1)$ -arithmetic from the two $g$ 's — each line system has its own "Runge denominator," which is why 1900s tables could catalog patterns by rational fractions before anyone knew what $j$ was.

Step 4. Scale for the exam. At $B=1$ T, $\mu_B B=5.79\times10^{-5}$ eV, so the D₂-type pattern spans $\tfrac{10}{3}\mu_B B=1.93\times10^{-4}$ eV — for sodium, resolvable interferometrically; for hydrogen Lyman-α, an $8\times10^{-7}$ fractional splitting in the vacuum ultraviolet. The pattern is identical; only the line center differs. (Sodium's absolute Zeeman energies at 1 T — 77.2 and 38.6 μeV per rung for the two upper levels — are Projection Theorem for Vector Operators' worked numbers, quoted not recomputed.) Verified: both sweeps executed programmatically (Δm ∈ {0,±1}): counts 4 and 6, shift sets {±2/3, ±4/3} and {±1/3, ±1, ±5/3}, zero-shift absence, all shifts k/3, and the g_u = g_l = 1 control collapsing to the 3-line triplet (article verify script).

Where weak-field theory ends

The premise was that the field only fans each $j$ -level without making neighbors talk. But $\hat{S}_z$ has matrix elements between $j$ -levels of the same $m_j$ — the projection theorem is silent about those, and they are the regime's undoing. For hydrogen's $2p$ pair the relevant element is $\langle 2p_{3/2},m_j|\hat{S}_z|2p_{1/2},m_j\rangle=-\tfrac{\sqrt2}{3}\hbar$ at $m_j=\tfrac12$ (built from the Clebsch-Gordan decomposition of Addition of Angular Momenta, evaluated exactly in the verify script), so the field couples the two fine-structure levels with strength $\tfrac{\sqrt2}{3}\mu_B B$ against their gap $\Delta E_{fs}=4.53\times10^{-5}$ eV.

Go deeper · Derivation C · 1 minThe breakdown field, computed as a mixing budget

Derivation CThe breakdown field, computed as a mixing budget

We quantify where "weak" ends by asking when the field-induced admixture of the neighboring $j$ -level reaches 10% in amplitude — a conventional but honest threshold beyond which the fan formula's errors turn visible. The strategy: first-order mixing estimate from the off-diagonal element, then an exact $2\times2$ diagonalization in the $m_j=\tfrac12$ sector as the audit. This derivation is the boundary marker; everything past it belongs to Strong-Field Paschen-Back Effect, whose intermediate-field machinery diagonalizes what we here only estimate.

Step 1. The coupling. In the $m_j=\tfrac12$ sector of $2p$ , the coupled states decompose as $|{\tfrac32},{\tfrac12}\rangle=\sqrt{\tfrac23}\,|m_\ell{=}0,\uparrow\rangle+\sqrt{\tfrac13}\,|m_\ell{=}1,\downarrow\rangle$ and $|{\tfrac12},{\tfrac12}\rangle=-\sqrt{\tfrac13}\,|0,\uparrow\rangle+\sqrt{\tfrac23}\,|1,\downarrow\rangle$ (Addition of Angular Momenta's table). Sandwiching $\hat{S}_z$ : $\langle{\tfrac32}|\hat{S}_z|{\tfrac12}\rangle=-\tfrac{\sqrt2}{3}\hbar$ (computed exactly). Only $\hat{S}_z$ contributes — $\hat{J}_z$ is diagonal in $j$ .

Step 2. Mixing estimate. First-order admixture amplitude $\approx\dfrac{(\sqrt2/3)\,\mu_B B}{\Delta E_{fs}}$ . Setting it to $0.1$ : $B_{10\%}=\dfrac{0.1\times3\,\Delta E_{fs}}{\sqrt2\,\mu_B}=0.17$ T (computed). One-sixth of a tesla — the fan formulas for hydrogen $n=2$ are percent-accurate only below a few hundredths of a tesla.

Step 3. Audit. Diagonalize the exact $2\times2$ ( $\hat H_{fs}+\hat H_Z$ in the $m_j=\tfrac12$ sector) at $B=0.17$ T: the eigenvector admixture comes out 0.092 — the estimate is honest to 10%. (The full intermediate-field treatment across all sectors, and its Breit-Rabi-style level curves, are Strong-Field Paschen-Back Effect's property.) Why sodium tolerates fields hydrogen cannot: the same √2/3 coupling divides by a fine-structure gap 47 times larger, pushing the 10% point to ~8 T. Regime boundaries are properties of the manifold, not of the formula.

Step 4. Reading the hierarchy. Weak-field validity is not one number but a ladder: percent-level fan accuracy, then visible rung curvature, then full $j$ -mixing, then Paschen-Back. Where you stand is always $\mu_B B/\Delta E_{fs}$ — the dimensionless dial installed by Zeeman Effect. Verified: the CG decomposition and the −√2ħ/3 element exact; B(10%) = 0.166 T; the 2×2 eigenvector admixture 0.092 at that field; the weak-window ratio μ_B B/ΔE_fs = 0.064 at 0.05 T (article verify script).

Limiting cases

case	result	meaning
$s=0$	$g_J=1$ : all fans equal, triplet restored	the normal Zeeman effect of Zeeman Effect as a special case (symbolic check)
$\ell=0$	$g_J=2$	pure spin fan — the moment dictionary's $g_s$ read directly off a spectrum
$j=\ell+\tfrac12$ vs $j=\ell-\tfrac12$	$g_J\gtrless1$	spin aligned vs anti-aligned with $\hat{\mathbf{J}}$ ; why $2p_{1/2}$ fans slower than orbital
$\ell\to\infty$	$g_J\to1$	spin's wage diluted by a long orbital vector; classical limit
$B\to B_{10\%}$ (0.17 T for H $n=2$ )	$j$ -mixing at 10%	fan formula degrading; intermediate machinery takes over
$\mu_B B\gg\Delta E_{fs}$	$m_\ell,m_s$ good	Paschen-Back regime — owned by Strong-Field Paschen-Back Effect

Common trapEqually spaced levels do not mean equally spaced lines

The fan theorem — each level splits into $2j+1$ rungs spaced exactly $g_J\mu_B B$ — tempts you to predict equally spaced spectral lines. Wrong: lines are differences between two fans with different $g_J$ , and the difference table is in general unevenly spaced and never contains an unshifted component when both levels carry spin (no solution of $g_um_u=g_lm_l$ with half-integer $m$ 's and unequal $g$ 's). The D₁-type census lands at $\pm\tfrac23,\pm\tfrac43$ — gaps of $\tfrac23,\tfrac43,\tfrac23$ , visibly uneven, with a hole where the triplet's central line would be. (The D₂-type set happens to come out evenly spaced in thirds — a coincidence of these particular $g$ 's, not a rule.) Draw both ladders and take differences; never shortcut from level spacing to line spacing. And the mirror error: do not use $m_\ell,m_s$ labels here — that basis belongs to the strong-field regime, where the fine structure has already lost.

Worked examples

Example 1 — the 2p₃/₂ quartet at 0.5 T, then the full Lyman-α pattern. Fan hydrogen's $2p_{3/2}$ level at $B=0.5$ T and predict the blue Lyman-α components.

Step 1. Spacing. $g_J\mu_B B=\tfrac43\times5.788\times10^{-5}\times0.5=3.86\times10^{-5}$ eV per rung.

Step 2. Rungs. $E^{(1)}=g_J\mu_B Bm_j$ for $m_j=+\tfrac32,+\tfrac12,-\tfrac12,-\tfrac32$ : $+5.79\times10^{-5}$ , $+1.93\times10^{-5}$ , $-1.93\times10^{-5}$ , $-5.79\times10^{-5}$ eV (all computed) — a symmetric quartet spanning $1.16\times10^{-4}$ eV, centroid unmoved.

Step 3. Add the ground fan: $1s_{1/2}$ splits to $\pm\mu_B Bm_j\times2=\pm2.89\times10^{-5}$ eV. Photon shifts from the sweep: $\{\pm\tfrac13,\pm1,\pm\tfrac53\}\times\mu_B B\times\tfrac{B}{1\,\text{T}}$ — at 0.5 T: $\pm0.96,\pm2.89,\pm4.82\times10^{-5}$ eV.

Step 4. Sanity: regime legal? $\mu_B B/\Delta E_{fs}=0.64$ at 0.5 T — marginal, past the 10% mixing point of Derivation C. Answer: quartet rungs $\pm1.93,\pm5.79\times10^{-5}$ eV; six-line pattern at $\pm0.96,\pm2.89,\pm4.82\times10^{-5}$ eV — with the honest caveat that at 0.5 T hydrogen $n=2$ already needs intermediate-field corrections at the several-percent level. (At 0.05 T every number divides by 10 and the fan is percent-exact.)

Example 2 — the sodium D census, the exam classic. How many Zeeman components do D₁ (589.6 nm, $3p_{1/2}\to3s_{1/2}$ ) and D₂ (589.0 nm, $3p_{3/2}\to3s_{1/2}$ ) each show in a weak field, and in what pattern?

Step 1. g-factors from (2): $g(3s_{1/2})=2$ , $g(3p_{1/2})=\tfrac23$ , $g(3p_{3/2})=\tfrac43$ — identical structure to hydrogen's Lyman system (alkali doublets are one-electron physics; the sodium rung energies at 1 T, 77.2/38.6 μeV, are quoted from Projection Theorem for Vector Operators).

Step 2. D₁ census: four components at $\pm\tfrac23,\pm\tfrac43\,\mu_B B$ (two π inside, two σ outside — swept).

Step 3. D₂ census: six components at $\pm\tfrac13,\pm1,\pm\tfrac53\,\mu_B B$ (π pair inside, σ quartet outside — swept).

Step 4. Regime honesty: sodium's crossover is 37 T (Zeeman Effect's computation), so at any tabletop field the census is exact — which is why these very photographs (Preston 1897) were sharp enough to defeat classical theory. Answer: D₁ → 4 components, D₂ → 6, all shifts thirds of $\mu_B B$ — the historical "anomalous" fingerprint, now two lines of $g_J$ arithmetic.

Origins · 1 minAnomalous Zeeman Effect and how this was discovered

Anomalous Zeeman Effect

1897–1925 · Dublin, Tübingen, Hamburg

The crisis began in Dublin: Thomas Preston's 1897 photographs showed the sodium D lines splitting into four and six components — patterns the Lorentz triplet theory could not produce at any parameter setting. Preston's rule (same series, same pattern) and Carl Runge's 1907 bookkeeping (all displacements rational fractions of the normal splitting) organized the zoo without explaining a single line of it. The decisive pattern-hunter was Alfred Landé at Tübingen, who in 1921 extracted the $g$ -formula — the exact expression this article consumes — by pure fitting, introducing half-integer quantum numbers because the data demanded them, years before anyone could say what they counted. The cost of living with the anomaly is preserved in Pauli's famous Hamburg reply to a colleague who asked why he looked so unhappy: "How can one look happy when he is thinking about the anomalous Zeeman effect?" The resolution arrived in 1925 from two directions at once: Pauli's exclusion principle needed a two-valued electron label, and Goudsmit and Uhlenbeck's spinning electron with $g_s=2$ turned Landé's fitted fractions into the projection arithmetic of the previous serials. By 1927 the "anomalous" effect was a homework problem — the word survives as a fossil of the twenty-eight years when it was instead the sharpest known crack in physics.

History

1897

— Preston photographs quartets and sextets in sodium; the classical triplet theory fails

1907

— Runge systematizes the patterns: all shifts are rational fractions of the normal splitting

1921

— Landé fits the g-formula from Zeeman data, introducing half-integer m before spin exists

1925

— Pauli's exclusion principle and the Goudsmit-Uhlenbeck spinning electron supply the missing label

1926

— Heisenberg and Jordan derive the anomalous patterns from quantum mechanics with spin

1927

— The projection-theorem derivation makes the vector model a theorem; the anomaly becomes arithmetic

Going further

Griffiths & Schroeter §7.4.1 — the weak-field case, with the same $g_J$ special cases.
Sakurai & Napolitano §5.3 — projection-theorem route, plus the intermediate-field discussion this article defers.
Cohen-Tannoudji, Diu & Laloë Vol. II, Complement $D_{XII}$ — anomalous patterns worked at full length.
Condon & Shortley, The Theory of Atomic Spectra, Ch. 16 — the classic pattern catalogs, Runge denominators and all.

Provenance note: Verifier should recompute: (1) the Landé formula via the projection theorem (Sakurai Ch. 3); (2) the special cases $g_J=1$ ( $s=0$ ), $g_J=2$ ( $\ell=0$ ); (3) the $^2P$ and $^2S$ g-factors $4/3,2/3,2$ (Griffiths Sec. 7.4, standard tables); (4) $\mu_B=9.274\times10^{-24}\,\mathrm{J/T}$ and $g_s\approx2.00232$ (CODATA). All numbers are standard textbook/CODATA values; symbol definitions match Griffiths, Sakurai, Shankar, Cohen-Tannoudji, Townsend. — Item (1) is executed where it is owned, in Projection Theorem for Vector Operators, whose derivation this article consumes by citation (its five special cases and sodium 77.2/38.6 μeV@1T anchors are quoted as its numbers); items (2)-(4) run in this article's verify script, together with the pattern censuses (4 and 6 components, shift sets in thirds), the quartet energies at 0.5 T, the CG-exact $j$ -mixing element $-\sqrt2\hbar/3$ , and the 0.17 T breakdown estimate audited by exact $2\times2$ diagonalization. Edition note: the base's own Landé-derivation steps are not reproduced here — the derivation belongs to the projection-theorem serial per chapter ownership; this article owns the weak-field application anatomy (fans, censuses, validity boundary).

Do I need to re-derive the Landé factor every time I use this formula?

No — that is the point of chapter ownership. The projection theorem and the $g_J$ derivation live in Projection Theorem for Vector Operators; this article consumes the result. On an exam, quote (2), compute the $g$ 's you need (seconds of arithmetic), and spend your time on what actually varies between problems: the fan diagram and the difference table under $\Delta m_j=0,\pm1$ .

Why is the quantization axis automatically along B — what if I pick another axis?

You may, but then $\hat{H}_Z$ is not diagonal in your $m_j$ and you will diagonalize your way back to the field axis. Choosing $\hat z\parallel\mathbf{B}$ makes $[\hat{H}_Z,\hat{J}_z]=0$ , which is what lets $m_j$ survive as a good label. The freedom you keep is the physics: with no field, all axes were equivalent — the $m$ -degeneracy of central potentials — and the field is precisely what breaks that tie.

The projection theorem replaces S with its component along J — where does the perpendicular part go?

Nowhere; it just averages to zero within a $j$ -multiplet. The perpendicular part of $\hat{\mathbf{S}}$ has matrix elements only between different $j$ values — the $-\sqrt2\hbar/3$ element of Derivation C. In a weak field those cross- $j$ elements lack energy leverage and are dropped; as $B$ grows they mix the fine-structure levels, and beyond the crossover they win entirely, handing the problem to the Paschen-Back basis.

Why does the pattern contain no unshifted line, when the normal triplet always has one?

An unshifted component needs $g_um_u=g_lm_l$ . With equal $g$ 's (singlets) the choice $m_u=m_l$ does it — the central π line. With unequal $g$ 's and half-integer $m$ 's there is no solution: every allowed pair shifts. A missing central line is therefore an immediate spectroscopic tell that both levels carry spin.

Is the anomalous pattern temperature- or intensity-dependent?

The positions are not — they are level structure, fixed by $g_J\mu_B B$ . The relative intensities of the components do depend on the thermal populations of the $m_j$ rungs and on geometry (π components vanish along $\mathbf{B}$ ), and full intensity formulas involve the squared dipole matrix elements between specific $(m_u,m_l)$ pairs. This article's censuses count components and place them; intensity bookkeeping belongs with the transition-rate machinery of the time-dependent chapter.

How does this generalize to hyperfine structure and the g_F factor?

By the same projection, one coupling deeper. Couple $\hat{\mathbf{J}}$ to nuclear spin $\hat{\mathbf{I}}$ into $\hat{\mathbf{F}}$ ; in a weak enough field the good states are $|F,m_F\rangle$ and the electron moment projects onto $\hat{\mathbf{F}}$ , giving $E=g_F\mu_B Bm_F$ with $g_F$ built from $g_J$ exactly as $g_J$ was built from $g_s$ and 1. The fields where that description breaks are a thousand times smaller (hyperfine gaps are ~GHz), which is why atomic-clock physics lives in the hyperfine version of this article — owned by Hyperfine Structure.

Landé fitted his formula in 1921 without knowing about spin — what exactly did he have?

Data and courage. He had Preston-rule pattern catalogs and Runge's rational fractions, and he found that assigning each level a factor $g=1+\frac{j(j+1)+s(s+1)-\ell(\ell+1)}{2j(j+1)}$ — with $j$ and $s$ taking half-integer values, scandalous at the time — reproduced every catalogued pattern. The formula was pure phenomenology: " $s$ " counted nothing anyone could name until 1925, when the spinning electron with $g_s=2$ made his fractions the projection arithmetic this chapter derives. It remains one of physics' great examples of an exact law extracted from data before its concepts existed.